Hi Balint,
I have contacted Ayim Tour - they are mentioned in LP guide book.
http://www.ayimtour.com/
Here's the tour description:
The description of this tour is:
Day 1
Depart hotel towards Aral Sea early in the morning. 9 hours drive will goes through following route Nukus - Kungrad - Plato Ustyurt - Sudochye Lake-Aral Sea. En route stop at Mizdakhan Mausoleum and Gaur-Kala Fortress. Arrival to Aral Sea is a little bit before the dusk. Driver will pitch a camp until you explore the surroundings. Enjoy your open air dinner. Overnight in Tent.
Day 2
You can shoot some great photos early in the morning at dawn. After breakfast visit to picturesque canyons of Ustyurt and drive towards Moynaq through the dried bottom of the sea. On arrival in Moynaq visit the graveyard of Ships in ex-shore of the Aral Sea and town museum. After visits drive to Nukus (210 km, 3 hours) and on arrival transfer to hotel for overnight. Overnight in Nukus.
The price of this tour is (for 4 person):
JEEP - $ 480/per car (four persons)
Tent – 1 (for two person) - $10/per two person
Sleeping bag - 1 - $3/per person
Meals - 1 - $36/per person (included: 1-lunch, 1-dinner, 1-breakfast)
Travel guide (if needed) - $50 per day
For the tour you will spend 1 and a half day.
Transport: total length of the road -900 km: JEEP- 480 USD (for 4 person)
I am afraid we cannot make it on May 8/9 since we are arriving to Urgench and going theclassic trip opposite way to Tashkent
BR,
Dmitry